reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th15:
  for f being homomorphic Function of Q1,Q2 st
  (for y be Element of Q2 holds ex x being Element of Q1 st f.x = y)
  &
  (for x,y,z be Element of Q1 holds a_op(x,y,z) in Ker f)
  holds
  Q2 is associative
proof
  let f be homomorphic Function of Q1,Q2;
  assume that A1: for y be Element of Q2 holds ex x being Element of Q1 st
    f.x = y
  and A2: for x,y,z be Element of Q1 holds a_op(x,y,z) in Ker f;
  thus Q2 is associative
  proof
    let x,y,z be Element of Q2;
    consider x1 being Element of Q1 such that A3: f.x1 = x  by A1;
    consider y1 being Element of Q1 such that A4: f.y1 = y by A1;
    consider z1 being Element of Q1 such that A5: f.z1 = z by A1;
    A6: a_op(x1,y1,z1) in Ker f by A2;
    a_op(x,y,z)= (f.x1 * f.(y1 * z1)) \ ((f.x1 * f.y1) * f.z1)
        by Def28b,A3,A4,A5
      .= f.(x1 * (y1 * z1)) \ ((f.x1 * f.y1) * f.z1) by Def28b
      .= f.(x1 * (y1 * z1)) \ (f.(x1 * y1) * f.z1) by Def28b
      .= f.(x1 * (y1 * z1)) \ f.((x1 * y1) * z1) by Def28b
      .= f.((x1 * (y1 * z1)) \ ((x1 * y1) * z1)) by Th13
      .= 1.Q2 by A6, Def29;
    hence thesis by Th9;
  end;
end;
